Fractions and Decimal Fractions

Fractions:

Any unit can be divided into any numbers of equal parts, one or more of this parts is called fraction of that unit. e.g. one-forth (1/4), one-third (1/3), three-seventh (3/7) etc.

The lower part indicates the number of equal parts into which the unit is divided, is called denominator. The upper part, which indicates the number of parts taken from the fraction is called the numerator. The numerator and the denominator of a fraction are called its terms.

A fraction is unity, when its numerator and denominator are equal.
A fraction is equal to zero if its numerator is zero.
The denominator of a fraction can never be zero.
The value of a fraction is not altered by multiplying or dividing the numerator and the denominator by the same number.e.g. 2/3 = 2/6 = 8/12 = (2/4)/(3/4)
When there is no common factor between numerator and denominator it is called in its lowest terms.e.g. 15/25 = 3/5
When a fraction is reduced to its lowest term, its numerator and denominator are prime to each other.
When the numerator and denominator are divided by its HCF, fraction reduces to its lowest term.

Proper fraction: A fraction in which numerator is less than the denominator. e.g. 1/4, 3/4, 11/12 etc.

Improper Fraction: A fraction in which numerator is equal to or more than the denominator. e.g. 5/4, 7/4, 13/12 etc.

Like fraction: Fractions in which denominators are same is called like fractions.

e.g. 1/12, 5/12, 7/12, 13/12 etc.

Unlike fraction: Fractions in which denominators are not same is called, unlike fractions.

e.g. 1/12, 5/7, 7/9 13/11 etc.

Compound Fraction: Fraction of a fraction is called a compound fraction.

e.g. 1/2 of 3/4 is a compound fraction.

Complex Fractions: Fractions in which numerator or denominator or both are fractions, are called complex fractions.

Continued fraction: Fraction that contain additional fraction is called continued fraction.

e.g.

Rule: To simplify a continued fraction, begin from the bottom and move upwards.

Decimal Fractions: Fractions in which denominators are 10 or multiples of 10 is called, decimal fractions. e.g. 1/10, 3/100, 2221/10000 etc.

Recurring Decimal: If in a decimal fraction a digit or a set of digits is repeated continuously, then such a number is called a recurring decimal. It is expressed by putting a dot or bar over the digits. e.g.

Pure recurring decimal: A decimal fraction in which all the figures after the decimal point is repeated is called a pure recurring decimal.

Mixed recurring decimal: A decimal fraction in which only some of the figures after the decimal point is repeated is called a mixed recurring decimal.

Conversion of recurring decimal into proper fraction:

CASE I: Pure recurring decimal

Write the repeated digit only once in the numerator and put as many nines as in the denominator as the number of repeating figures. e.g.

CASE II: Mixed recurring decimal

In the numerator, take the difference between the number formed by all the digits after the decimal point and that formed by the digits which are not repeated. In the denominator, take the number formed as many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits. e.g.